Thursday, December 8, 2016

Crossover Basics - The Zobel

Introduction to the Zobel

A Zobel network is used to flatten a driver's impedance (usually a woofer or mid-range), therefore making the filter (usually low-pass) more effective. A Zobel consists of a capacitor and resistor which are wired in parallel to a driver.

C1,R1 on the left are an example of the Zobel network. Their values are chosen to minimize the impedance rise of a driver above the resonant frequency.

Speaker Impedance

Introduction to Series Circuits

Take a look at a very simple series circuit. It consists of two resistors in series (one after another) with the amplifier. The circuit is closed by the ground points (the downward facing triangles).

As before, this may be something you like to play with, so I encourage you to grab a copy of XSim and try this out along with the Peak Voltage chart.

What's important to understanding here is that the voltage across the resistors will be proportional to the resistance offered. So

Vr1 = Vin * R1/(R1 + R2)
Vr2 = Vin * R2/(R1 + R2)

And of course:

Vr1 + Vr2 = Vin

That is, the voltage across R1 and R2 must add up to the input voltage.

Think of Ohms as elephants that eat volts. The more elephants, the larger portion of the incoming voltage they eat. Yes, this is a very silly analogy. Still, we can do some quick math. The total resistance is 100 Elephants (hah!) or 100 Ohms. R1 has only 10 elephants, so it gets 10% of the incoming voltage, whatever that voltage may be. R2 has 90% of the Elephants, so it takes 90% of the incoming voltage. There's a lot more to circuit analysis, but this is the bare minimum to understanding Zobel networks. Hopefully you'll be intrigued and learn more on your own.

Woofer Impedance

So with a little background under your belt you are now ready to look at a typical woofer. We'll use the LM-1 woofer, the Peerless 830991. You should know that impedance graphs will change once a driver is in the cabinet, especially if the cabinet is ported, so the data I present here will be different than a specification sheet which measures the driver in free-air.

You have heard the term "coil" used interchangeably with
"inductor." Which is correct, but you may not have thought about the
term "voice coil" in the same context. The voice coil is the part of the
speaker that will electrically connect to your amplifier and produces
the magnetic force which moves it against the magnetic field of the
permanent magnet. We won't get too much into this, but suffice it to say
that above the resonant peak the voice coil behaves like that of any
other inductor, specifically it has both a resistive element (DC
Resistance, or Re) and an inductive element (Le). These combine to give
us the woofer's electrical impedance (Z) at any given frequency.

In the chart below we will compare the impedance of the woofer in a sealed cabinet (red) with a woofer that has a Zobel network applied (blue).

Let's ignore what happens below 200 Hz. That's a topic beyond this posting, and it's also far below our likely filter points. From about 200 Hz to 400 Hz the woofer is purely resistant, and we have the minimum around 6.6 Ohms, but what happens to the right? That's correct, suddenly more voltage-eating elephants arrive! The inductive qualities of the voice coil become more and more important and overwhelm the well behaved resistance. Compare the peak difference of the two impedance charts, all the way at the right. The blue line represents a woofer compensated with a Zobel network. The impedance never goes above 7 Ohms, while the normal un-compensated woofer goes to over 30! That's more than 4:1 difference. This increase impedance is going to compete with the low pass filter and make it behave in ways we probably don't want.

For clarity we'll leave behind the LM-1 schematic and create a new one, with two identical woofers and 2nd-order low-pass filters set to 4 kHz. Of course, this is not how a real speaker would be designed, we are just using this to see exactly how a Zobel circuit works. The Zobel consists of C1 and R1.

The first thing we should do is examine the transfer function of the two filter sections:

As you can see, S2 is behaving like we expect a low pass filter to work. S1 however is having a very difficult time getting to the right slope. At 5kHz the output is almost 10 dB higher than we want it to be. That's a big deal. Eventually the impedance (elephants) on the low pas filter take over, but they don't reach our desired behavior until past 20 kHz, definitely not good enough for us. Let's take a look at the final outcome, below:

How Does a Zobel Work?

Above we discussed how serial components work in a circuit. You may feel a little tricked because while we learned enough to understand why coil inductance needs to be compensated for, we never talked about how a parallel circuit works, which is what a Zobel is. A parallel circuit has one unique property:

The apparent impedance of a parallel section is never more than the smallest impedance.

If we imagine C1 as a short, then no matter what S2 rises to, the impedance will never go above R1, or 8.2 Ohms. It can be less than that, but never more. In a parallel circuit you calculate the apparent impedance like so:

Rtotal = 1 / ( (1/R1) + (1/R2) + (and so on and so forth) )

Things are more complicated because we are actually calculating impedance, but you get the picture.

Let's do some quick, Dr. Leach style of analysis on the components in a Zobel. At very low frequencies, C1 behaves like an open circuit, essentially removing R1 and C1 from meaningful contributions to the system impedance. Remember we mentioned that impedance cannot rise more than the smallest value? So at low frequencies C1 is so large that S2 becomes the limit on impedance. You can see the impedance below 500 Hz or so is barely affected. At high frequencies, C1 rapidly decreases until it evectively becomes a short, putting R1 in parallel with the driver, S2 and limiting the absolute maximum impedance to 8.2 Ohms. In this case we don't reach 8.2 until well-past 20kHz but it would eventually reach that point once the woofer's impedance was high enough. Also past our point of concern. If we can limit the impedance from 6.6 Ohms to 7 Ohms then we have a much more stable impedance curve than before, and that's good enough.

Do I Need a Zobel?

That's a tricky question! So let's examine this woofer and it's output. If you wanted to cross it over at 4kHz I would think the Zobel was mandatory, however if you were going to set your crossover frequency at 2kHz or lower I would say not really. The LM-1 uses it, but the effect of the Zobel is small, and benefits the phase response so I leave it in. It is possible a very similar sounding LM-1 could be built without a Zobel and with different choices in the filters left behind. The best chart to look at to see if a Zobel matters in your circuit is the transfer function chart.

It is very rare, but not unheard of, that a tweeter needs a Zobel because their voice coils are relatively tiny and therefore don't have a lot of inductance. The most common exception to this rule is with ribbon tweeters. The ribbon itself is not inductive but the entire assembly often include matching transformers. Transformers are coils .... and coils are inductive... see where this goes? :)

The real point to the Zobel is to make things better in the area you need the filter to behave at it's best. That's usually up to about -20 to -30 dB. Beyond that if your slope isn't perfect we no longer really care. There's no audible difference between -60 and -67 dB for example.

An important consideration in choosing a Zobel or not is that they are not free. The more parts in a system, the more expensive, the more chances of failures or parts being out of specification. If this is a personal project, no problem, it's all experience. However if you are building for mass production eliminating unnecessary components is the final stage before committing a design to the factory.

Placement

In most cases, you want to place a Zobel closest to the driver. Put anything else such as padding resistors, filters, etc. before it. While the order of serial components does not matter, the order of parallel components does. Leaving the Zobel last prevents unexpected consequences.

There is a rare exception, when you must equalize a driver (usually a tweeter) by adding inductance. In which case you want the equalizing circuit closest to the tweeter, then the Zobel, so the Zobel can also control the EQ's impedance. I'll write more about this later in a section on handling difficult tweeters.

The Secret Uses of the Zobel

Many will rely on on-line calculators to determine the right values for a Zobel network, and that's fine, but be aware that the absolute values can be tweaked. The main benefit of this tweaking is to gently nudge the phase charts one way or the other, helping you get near-perfect matching between two drivers you otherwise might not have. This is where having a tool like Xsim to simulate your tweaks comes in super handy.

Exercises

The Peerless 830991 has an Re of around 6.6 and Le of around 0.330mH. Try simulating this in Xsim using a resistor and coil in series. Compare your impedance curve with the red impedance curve, above. What's the biggest difference you see?

Try using an online-calculator to create a Zobel for this driver. Tweak the capacitor and resistor values. Can you do better than the on-line calculator?

Using the complete LM-1 schematics, compare the woofer response with and without the Zobel. Is it a big difference? Can you fix the LM-1 so it no longer needs a Zobel? What difficulties did you encounter?Pay attention to the phase matching as well as the frequency response.